A mean-square bound concerning the lattice discrepancy of a torus in ℝ³

Abstract

For positive constants a > b > 0, let P _T (t) denote the lattice point discrepancy of the body tT _a,b , where t is a large real parameter and T = T _a,b is bounded by the surface

$$ \partial \tau _{a,b} :\left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {(a + b\cos \alpha )\cos \beta } \\ {(a + b\cos \alpha )\sin \beta } \\ {b\sin \alpha } \\ \end{array} } \right), 0 \leqq \alpha ,\beta < 2\pi . $$

In a previous paper [12] it has been proved that

$$ P_\tau (t) = \mathcal{F}_{a,b} (t)t^{3/2} + \Delta _\tau (t), $$

where F _a,b (t) is an explicit continuous periodic function, and the remainder satisfies the (“pointwise”) estimate Δ _T (t) ≪ t ^11/8+ɛ. Here it will be shown that this error term is only ≪ t ^1+ɛ in mean-square, i.e., that

$$ \int\limits_0^T {(\Delta _\tau (t))^2 dt} \ll T^{3 + \varepsilon } $$

for any ɛ > 0.